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Today's calculation of Integral 528

Source: 2002 Tokyo University entrance exam/Science I, 2nd exam

January 13, 2010
calculusintegrationfunctiongeometrygeometric transformationrotationlimit

Problem Statement

Consider a function f(x)\equal{}xe^{\minus{}x^3} defined on any real numbers. (1) Examine the variation and convexity of f(x) f(x) to draw the garph of f(x) f(x). (2) For a positive number C C, let D1 D_1 be the region bounded by y\equal{}f(x), the x x-axis and x\equal{}C. Denote V1(C) V_1(C) the volume obtained by rotation of D1 D_1 about the x x-axis. Find lim⁡C→∞V1(C) \lim_{C\rightarrow \infty} V_1(C). (3) Let M M be the maximum value of y\equal{}f(x) for x≥0 x\geq 0. Denote D2 D_2 the region bounded by y\equal{}f(x), the y y-axis and y\equal{}M. Find the volume V2 V_2 obtained by rotation of D2 D_2 about the y y-axis.
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